Studying Galaxy-Galaxy Lensing and Higher-Order Galaxy-Mass Correlations Using the Halo Model

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Studying Galaxy-Galaxy Lensing and Higher-Order Galaxy-Mass

Correlations Using the Halo Model

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.) der

Mathematisch-Naturwissenschaftlichen Fakult¨ at der

Rheinischen Friedrich-Wilhelms-Universit¨ at Bonn

vorgelegt von Jens R¨ odiger

aus Dortmund

Bonn 2009

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http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. Das Erscheinungsjahr ist 2009.

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨ at der Rheinischen Friedrich-Wilhelms-Universit¨ at Bonn

1. Referent: Prof. Dr. Peter Schneider

2. Referent: Prof. Dr. Cristiano Porciani

Tag der Promotion: 8.6.2009

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Introduction and Overview

In recent years, our understanding of the evolution of the Universe has made a great leap forward. A decisive factor for this achievement is that we have collected an enormous amount of high-quality cosmological data, e.g., the positions and shapes of millions of galaxies. The improvements in data collection are founded in the design of modern observational instruments and the ability to store and process large amounts of data on computers. Along with the progress of the experimental techniques, we have developed a consistent theoretical framework which accurately reproduces the data. This (ongoing) progress makes cosmology a precision science where important cosmological parameters can be constrained with percentage accuracy. However, our improved understanding also raises new fundamental questions of the physical processes involved. Most importantly, the nature of dark matter and dark energy, which together comprise 96 per cent of the energy composition of the Universe, is still unknown. Dark matter is postulated to explain, among other observations, the flat rotation curves in spiral galaxies. Up to now, attempts to directly detect dark matter particles or to reproduce them in high-energy accelerators have failed. Dark energy is a hypothetical form of energy which is introduced to explain the current accelerated phase in the expansion history of the Universe. At the moment and in the near future several complementary experiments are underway and being planned to get an insight into these unsolved issues, making this a particular exciting time for cosmologists.

A successful cosmological model needs to explain measurements of the early Universe, such as the abundance of primordial elements and the temperature fluctuations in the cosmic microwave background (CMB) radiation, as well as measurements of the local Universe such as the distribution of galaxies. The connection between the two regimes is that the large-scale structure we observe today is believed to have formed by gravitational collapse of small density fluctuations that were present in the early Universe. With the knowledge of the initial distribution of these density perturbations, which can be accurately measured by observations of the CMB temperature fluctuations, we can predict the statistical properties of the large-scale structure observed today. To describe this evolution, we need the physics of gravitational clustering as described by general relativity or for scales much smaller than the Hubble radius even by a Newtonian approach. Up to now, theoretical predictions have been essentially limited to dark matter fluctuations, whereas observations measure the light or galaxy distribution of the Universe. The difficulty is that, in general, the statistics of galaxy clustering is not the same as the statistics of dark matter clustering and the scale-dependent difference between both is known as the galaxy bias. To find a connection between theoretical predictions and observations, the modeling of the bias is one of the crucial challenges in

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modern cosmology.

A promising cosmological probe to infer the bias is galaxy-galaxy lensing which describes the deflection of light from background galaxies caused by the gravitational field of foreground galaxies. The gravitational field around the foreground galaxies is dominated by the dark matter halos in which the galaxies are embedded. The advantage of galaxy-galaxy lensing compared to other cosmological probes is that it does not rely on luminous tracers of the underlying mass distribution. Moreover, it can probe the potential of the dark matter halo out to much larger distances from the halo center than it is possible with measurements of rotation curves in spiral galaxies. Recently, the concept of galaxy-galaxy lensing has been generalized to a method which is sensitive to the distortion pattern around pairs of foreground galaxies rather than the distortion around a single galaxy. This new method is termed galaxy-galaxy-galaxy lensing (GGGL). A potentially beneficial application of GGGL is to study the environment of bound systems which are composed of a small number of galaxies, like galaxy groups.

Theoretical predictions of measurements of galaxy-galaxy lensing need to provide an accurate model of gravitational clustering. These models necessarily have to include a treatment of nonlinearities in the matter density field in order to describe the small-scale regime of the measurements. There are three main approaches to deal with these nonlinearities: on large to intermediate scales, the dynamical equations can be solved analytically with a perturbative ansatz which, however, breaks down on small scales.

Alternatively, one can simulate the evolution of the density and velocity fields in dark matter N-body simulations. The drawback of using simulations is that they are limited to a specific volume size and are very time-consuming to conduct. Finally, there are analytic models which combine the results from simulations and theoretical results.

These models allow for a physical interpretation and may help to find a solution of the gravitational clustering equations valid for the whole range of scales. The drawback is that they need to be well tested against numerical simulations.

We apply the third approach and consider an analytic model where all the dark matter of the Universe is bound in spherically symmetric halos. The standard paradigm for the formation of galaxies is then that baryonic gas can only cool and form stars in potential wells which are provided by dark matter halos. On large scales, galaxy clustering is then dominated by the well-known clustering of halos, and on small scales it is dominated by the clustering of galaxies in their host halo. The latter can be predicted by modeling the halo occupation distribution which is the mean number of galaxies contained in a halo of a specific mass. In addition, one needs to specify the radial distribution of galaxies in their host halo. This analytic approach is known as the halo model for galaxy clustering. If one considers only dark matter clustering one speaks of the dark matter halo model.

To extract cosmological information from the observed large-scale structure, the best we can do is to adopt a statistical approach where our observable Universe is a stochastic realization of a random field. The key observables are the moments of this random field which are the n-point correlation functions in real space. Only for the special case of Gaussian random fields does the two-point correlation function, or its

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Fourier counterpart the power spectrum, encode the full statistical information of the field. However, the process of structure formation inevitably leads to nonlinearities in the fields which also give rise to higher-order correlation functions. It is interesting to study the three-point correlation function, or its Fourier counterpart the bispectrum, since it is the lowest-order non-vanishing moment which describes non-Gaussian effects.

Furthermore, it is beneficial to study the fourth-order moment, which is the so-called trispectrum in Fourier space, since it determines the expected statistical errors for a given power spectrum estimator. In addition to these aspects, the determination of higher-order spectra allows one to lift cosmological parameter degeneracies and to enhance the signal-to-noise ratio of observations of galaxy clustering. The halo model provides a simple framework for analytic calculations of higher-order spectra.

Overview

In this thesis we focus on the modeling and cosmological interpretation of higher-order spectra. In particular, we aim to develop a quantitative model for the GGGL signal, combining the dark matter halo model and the halo model for galaxy clustering. In addition, we want to predict the statistical error matrix for a given unbiased estimator for the projected matter-galaxy power spectrum which is applicable for the whole range of scales probed by observations. The results can be used to perform a likelihood analysis of the galaxy-galaxy lensing signal which shows how well potential future experiments can constrain cosmological parameters.

The outline of the thesis is as follows:

• In Chapter 1, we derive the important relations of the homogeneous background Universe and discuss how different cosmological probes can determine the important cosmological parameters. In addition, we present the observational evidence which in recent years has led to the cosmological standard model. As an important example for a mechanism beyond the standard model, we discuss the inflationary phase of the early Universe.

• Chapter 2 deals with the physical description of cold dark matter structure formation via the nonrelativistic fluid equations. We show that these can be solved with a perturbative approach, first presenting the well-known linear solution, and then giving a valid general perturbative solution. In addition, we introducen-point correlation functions which are used to infer statistical information on the matter or galaxy clustering.

• In Chapter 3, we first present the ingredients of the dark matter halo model such as the halo mass function, the halo density profile and the halo bias. Then we show that we can construct general n-point correlation functions in terms of these ingredients which are valid on large and on small scales. We give explicit results for the two-, three- and four-point correlation functions and their corresponding

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Fourier space counterparts, i.e., the power spectrum, the bispectrum and the trispectrum, which are needed for the subsequent chapters.

• The results of the dark matter halo model are extended to the halo model of galaxy clustering which is shown in Chapter 4. The main new ingredients are the halo occupation distribution P(N|m) which is the conditional probability that a halo of mass m containsN galaxies, and the radial distribution of galaxies in their host halo. We focus on the development of cross-spectra which are probed by galaxy-galaxy lensing, and give the explicit relations for the power spectra, bispectra and trispectra.

• In Chapter 5, we review the basic concepts of gravitational lensing. Then we focus on cosmic shear as a cosmological probe whose signal is a filtered version of the angular spectra. The angular and spatial spectra are related by Limber’s approximation. We use our implementation of the dark matter halo model developed in Chapter 3 to produce theoretical predictions.

• In Chapter 6, we first discuss galaxy-galaxy lensing and the estimation of its signal with the halo model. The main emphasis is on the recently introduced GGGL method for which we show halo model predictions of the signal. For these predictions we need the results of the halo model for galaxy clustering as given in Chapter 4.

• Chapter 7 deals with the theoretical modeling of the covariance of the galaxy- galaxy lensing power spectrum. In particular, we include the non-Gaussian part which was neglected in previous studies. Moreover, we analyze the influence of shot and shape noise on the correlations of different scales.

The main new results of this thesis are summarized at the end of Chapter 6 and Chapter 7. The thesis concludes with a general summary and gives an outlook on future related work.

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Chapter 1 The Cosmological Standard Model

At first sight it seems impossible to describe the evolution of the Universe. Fortunately, observational data indicate that on the largest scales we live in a highly symmetric Universe which makes a theoretical description feasible. Going to smaller scales, however, we see many structures like galaxies that are not distributed isotropically. The standard paradigm is that these structures originated from small perturbations that were present in the early Universe and which subsequently grew due to gravitational instability. Here we focus first on the cosmological model of the homogeneous background Universe neglecting all perturbations. In the next chapters we will discuss in detail the process of structure formation leading to the observed inhomogeneous structures on small scales in the environment of the mean background model. Understanding the properties of the mean background Universe and the dynamical formation of structure are the key theoretical concepts that are needed for the interpretation of experimental results of cosmological probes.

The great improvement of observational techniques, theoretical models and numerical simulations in recent years helped us to develop a consistent picture of the origin and evolution of the Universe, the so-called standard model. It is based on the following observations: the expansion of the Universe first observed by Hubble, the observation of primordial elements from nucleosynthesis formed in the early Universe and the isotropic temperature distribution of the photons of the cosmic microwave background (CMB) radiation. However, there are still many open questions, for example the origin of the accelerated expansion in the recent history of the Universe, the (particle) nature of dark matter and the physical process that led to inflation in the early Universe. The advent of precision cosmology, which allows us to determine cosmological parameters at percentage-level accuracy, will constrain the concordance model even better with upcoming surveys. Maybe, taking advantage of the present rapid theoretical and observational progress, we will then be able to probe the processes that are not (yet) part of the standard model.

This chapter is organized as follows: In Sect. 1.1 we introduce the concept of a homogeneous background Universe and derive the most important relations which describe the kinematical and dynamical properties of this background Universe. We review then the determination of the cosmological density parameters in Sect. 1.2 that quantify the energy budget of the Universe. This is followed by Sect. 1.3, where we

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give a concise presentation of the physical phenomena that are part of the cosmological standard model and review the history of the Universe from the Big Bang to the current large-scale structure. Furthermore, we discuss the inflationary phase of the very early Universe which is a necessary extension of the standard model.

1.1 Homogeneous Background Universe

In this section we derive the most important relations of the homogeneous background Universe which are used for the rest of the thesis, namely the Friedmann equations, the cosmological redshift and the distance-redshift relations.

1.1.1 Robertson-Walker Metric

TheCopernican principle¹ states that the Earth has no special position in the Universe.

A generalization of this concept leads to the assumption that there is no favored position in space at all. In addition, observations indicate that the Universe looks isotropic around the Earth. Combining both points, we deduce isotropy around every point in space. One can show mathematically that global isotropy implies homogeneity of space.

In summary, this leads to the famous statements 1. the Universe is homogeneous,

2. the Universe is isotropic,

which is the so-calledcosmological principle. Of course, the homogeneity and isotropy of the Universe does not apply on small scales because we see a large variety of galactic and extragalactic structures. It is rather understood as an averaging process over sufficiently large cells with spatial extent of >100 Mpc.

When applying the cosmological principle, Robertson and Walker independently showed that the general metric of space-time ds² ≡g_µνdx^µdx^ν can be reduced due to the underlying symmetry to the simple form of²

ds² =c²dt²−a²(t)

dw²+f_K²(w)dΩ²

, (1.1)

with

dΩ² = dθ²+ sin²θdφ². (1.2)

These are the coordinates offundamental observers which move on geodesics with the cosmic fluid. The spatial coordinates of these fundamental observers are thus called comoving coordinates here denoted by (w, θ, φ) and t is the cosmic time as measured in

1This goes back to the work of Nicolaus Copernicus who showed that the Earth is not the center of the solar system but is instead moving around the Sun.

2We assume the Einstein summation convention, namely that we sum over identical upper and lower indices. Greek indices run over the four relativistic coordinates, e.g.,µ= 0. . .3.

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the rest frame of a comoving clock. We see that the evolution of the Universe is governed by the time-dependent scale factor a(t). The form of the metric shows that space-time can be decomposed into spatial hypersurfaces of constant time and curvature, where isotropy requires these three-dimensional surfaces to be spherically symmetric. The comoving angular diameter distance f_K(w) is either a trigonometric, linear or hyperbolic function of w corresponding to the curvature K of the hypersurfaces:

f_K(w) =







K^−1/2sin(K^1/2w) for K >0,

w for K = 0,

(−K)^−1/2sinh[(−K)^1/2w] for K <0.

(1.3)

Note that in this convention of the Robertson-Walker metric the curvature K has the units of length⁻² and the scale factor is dimensionless.

1.1.2 Cosmological Redshift

Edwin Hubble discovered in 1929 that almost all galaxies are receding from us with a radial velocity that is on average proportional to their distance from us. As a result of the expansion of the Universe, the light of distant sources arriving at the Earth appears redshifted in its wavelength. Let us consider a light ray that is emitted from a comoving source at time t_e (“e” for emitter) and reaches a comoving observer at time t_o (“o” for observer). Since light rays travel on null geodesics we have ds² = 0. In addition, we assume that we have radial light rays which are characterized by a constant value of θ and φ. If we apply these conditions to the metric (1.1) we find

c|dt|=a(t)dw . (1.4)

The coordinate distance for both comoving observers remains constant by definition:

w_eo= Z to

te

dw= Z to

te

dt c

a(t) = const. (1.5)

This equation can be written equivalently as dt_o

dt_e = a(t_o)

a(t_e). (1.6)

Describing the infinitesimal change in time by dt≡ν⁻¹, where ν is the frequency of the light wave, Eq. (1.6) becomes

ν_e ν_o = λ_o

λ_e = 1 + λ_o−λ_e

λ_e = 1 +z = a(t_o)

a(t_e), (1.7)

where the introduced redshift factor z describes the relative change in the wavelengthλ.

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For the rest of this work we deal with the situation where the observer is situated on Earth or in space and the time of the observer is today. The scale factor is normalized to unity today, i.e., a(t0) = 1. Then the relation between the redshift z and the scale factor a is

a(z) = 1

1 +z . (1.8)

1.1.3 Einstein Field Equations

In the previous sections we applied the cosmological principle to achieve a geometrical description of the Universe. In the following, we want to predict the time dependence of the scale factor and therefore the dynamics of the Universe. The fundamental equations used to describe the dynamics of the Universe are the Einstein field equations. In an enormous intellectual work, Einstein developed an extension of his theory of special relativity which incorporates also gravity. This theory, known as general relativity, describes space-time as a four-dimensional manifold with a metric tensor g_µν. The Einstein equations couple the metric with the matter-energy content of the Universe that is described by the energy-momentum tensor Tµν:

G_µν =−8πGN

c⁴ T_µν, (1.9)

with the Einstein tensor given by

G_µν =R_µν −1

2Rg_µν+ Λg_µν, (1.10)

where R_µν is the Ricci tensor and its contraction R is called the Ricci scalar. Both depend on first- and second-order derivatives of the metric. The Einstein tensor and the energy momentum tensor satisfy the Bianchi identity ∇^νGµν = 0 and energy conservation ∇^νT_µν = 0, respectively. Here∇^ν denotes the covariant derivative. Since the metric is constant with respect to covariant derivatives (∇^αg_µν = 0), the stated conservation laws allow to add a term that is proportional to the metric on either side of the Einstein equation. This was introduced as the Λ-term by Einstein in 1917 on the left-hand side of Eq. (1.9) because he wanted to achieve a static Universe, i.e., a Universe which is not expanding at all. Alternatively, one can also put the Λ-term on the right-hand side of Einstein’s equation. Then it acts as a source of energy as Heisenberg already pointed out. In this case theoretical particle physics tells us that the constant term could be the vacuum energy of the Universe. We will come back to this point later in this section. Note that we already included the Λ-term in the Einstein tensor in Eq. (1.10). Taking the weak-field limit of the Einstein equation including the Λ-term leads to a modification of the Poisson equation of Newtonian gravity which is discussed in detail in Sect. 2.1.1.

The energy-momentum tensor of the Universe is assumed to be that of a perfect relativistic fluid

T_µν = (ρ+p/c²)U_µU_ν −pg_µν, (1.11)

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where ρ is the density andpis the pressure of the perfect fluid and U_µ is the relativistic four-velocity. For comoving coordinates the four-velocity takes the particular simple form of U^µ = (1,0,0,0).

1.1.4 Friedmann Equations

The Einstein equations (1.9) can be solved for a given metric. Choosing the Robertson- Walker metric (1.1), we obtain the two famous Friedmann equations for a homogeneous and isotropic Universe:

H²(t)≡ a˙

a 2

= 8πG_N

3 ρ− Kc² a² +Λ

3 , (1.12)

¨ a

a =−4πG_N 3

ρ+ 3p

c²

+ Λ

3 , (1.13)

where we have also introduced the Hubble rate H(t). The value of the Hubble rate today is the so-called Hubble constant

H₀ = 100hkm s⁻¹Mpc⁻¹, (1.14)

where the Hubble parameter h quantifies the uncertainties in the measurements. The origin of this parameter is founded in the large uncertainties in the measurements of the Hubble constant several years ago. Today the Hubble key project provides a much more accurate determination of the Hubble constant (Freedman et al. 2001), where they find h = 0.72±0.08. We note that we can define a critical point from the first Friedmann equation(1.12): the curvature of the Universe is flat, implying K = 0, if the density is equal to the critical density

ρ_crit(t) = 3H²(t)

8πG_N , (1.15)

where we neglected the Λ-term in the first Friedmann equation. It is possible to combine both Friedmann equations to the adiabatic equation

d

dt(a³ρc²) +pd

dta³ = 0, (1.16)

which is analog to the first law of thermodynamics. In other words, the change in energy in a comoving volume element is equal to minus the pressure times the change in volume.

Note that it is possible to find this equation, without solving the Einstein equations, from the energy-momentum conservation of general relativity, i.e., ∇µT^µν = 0.

We now derive the time dependence or alternatively the dependence on the scale factor of the density ρ by solving the adiabatic equation (1.16). However, we need a relation between the pressure and the density. This is obtained by an equation of state (EOS) similar to the ideal gas in thermodynamics. We parametrize the EOS by

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p=p(ρ) =c²wρ, where wis the equation of state parameter. The most general case we consider is a time-dependent w, in which case the solution of the adiabatic equation is

ρ(a) = ρ₀exp

−3 Z a

1

da⁰

a⁰ [1 +w(a⁰)]

, (1.17)

where ρ0 is the density of the considered matter species today, i.e., ρ0 ≡ρ(a= 1). If w is constant then the integration in the exponential can be performed and yields

ρ(a) =ρ₀a^−3(1+w). (1.18)

This implies that to find the parameter w, we must specify the equation of state for certain forms of matter. The matter content in the Universe can be divided into relativistic and non-relativistic matter which are often called radiation and pressureless dust, respectively. For radiation the EOS is derived from special relativity as

pr= ρ_rc²

3 , wr= 1

3. (1.19)

Non-relativistic matter can be approximated by

p_m= 0, w_m = 0. (1.20)

Using these relations, we find from Eq. (1.18)

ρ_m(t) = ρ_m,0a(t)⁻³, ρ_r(t) = ρ_r,0a(t)⁻⁴, (1.21) for matter and radiation density, respectively. These results provide an intuitive interpretation: The first equation describes the dilution of the number density of particles with the expanding Universe. Radiation has an additional reductional factor of a⁻¹ due to the energy dependence on redshift.

In 1998, the accelerated expansion of the present Universe was independently detected by two groups observing supernova type Ia (SNIa) (Riess et al. 1998; Perlmutter et al.

1999). This came as a surprise because the physical origin of the accelerated expansion is still unknown and one of the biggest theoretical and observational challenges in modern physics. The mysterious form of energy that is responsible for the acceleration is called dark energy. From the second Friedmann equation (1.13) an accelerated expansion (¨a(t)>0) occurs when the equation of state parameter fulfills w_de<−1/3.

In particular, a constant energy density can be achieved when w_de= −1 (see Eq. 1.18).

This is the simplest dark energy model and is calledcosmological constant. When the EOS parameter is time dependent, one needs to use the general equation (1.17) and we speak of quintessence models of dark energy.

Introducing the density parameters Ω_i ≡ρ_i,0/ρ_crit,0 for the i-th species and replacing ρ→ρr+ρm =ρm,0a⁻³+ρr,0a⁻⁴, we can rewrite the first Friedmann equation (1.12) as

H²(a) = H₀²

a⁻⁴Ω_r+a⁻³Ω_m−a⁻²Kc² H₀² + Ω_Λ

, (1.22)

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where the density parameter for the cosmological constant is defined by Ω_Λ ≡ Λ

3H₀² . (1.23)

Setting t=t₀ in Eq. (1.22) the parameters today are related by K =

H₀ c

2

(Ω_m+ Ω_r+ Ω_Λ−1)≡ H₀

c 2

(Ω_tot−1), (1.24) where we defined Ω_tot in the second step, which is the total density parameter today.

This equation tells us that the total energy density and the geometry of the Universe are closely related: for Ω_tot = 1 we obtain a flat Universe (K = 0), Ω_tot <1 corresponds to an open Universe (K <0) and Ω_tot >1 characterizes a closed Universe (K >0).

Replacing the curvature in Eq. (1.22)with this relation, the first Friedmann equation finally reads for the matter-dominated era, i.e., for Ω_r 1:

H²(a) =H₀²

a⁻³Ω_m+a⁻²(1−Ω_m−Ω_Λ) + Ω_Λ

. (1.25)

Scale-factor dependent density parameters Ω_i(a) can also be defined. Their evolution can be written explicitly by using Eq. (1.25):

Ω_m(a)≡ ρ_m(a)

ρ_crit(a) = 8πG_N

3H²(a)ρ_m,0a⁻³ = Ω_m

a+ Ω_m(1−a) + Ω_Λ(a³−a), (1.26) Ω_Λ(a) = Λ

3H²(a) = Ω_Λa³

a+ Ω_m(1−a) + Ω_Λ(a³−a), (1.27) where we also neglected the contribution from Ω_r. These equations have the asymptotic behavior Ω_m(a)→1 and Ω_Λ(a)→0 as a →0 independent of the values of Ω_m and Ω_Λ today.

1.1.5 Cosmological Distance Measures

Here we introduce the main distance measures used in cosmology and derive their distance-redshift relations.

The comoving distance Dcom is the distance between two observers comoving with the cosmic flow, thus D_com ≡w. We found in Sect. 1.1.2 that radial light rays fulfill cdt =−adw. We rewrite this relation in terms of the scale factor

dw=−c

adt=−c a

da dt

⁻¹

da=− c

a²H(a)da , (1.28) where we used the definition of the Hubble rate in Eq. (1.12) in the last step. Thus, the comoving distance for an observer at a= 1 to a source located at the scale factor a is

w(a) = c H₀

Z 1 a

da⁰[a⁰Ω_m+a⁰²(1−Ω_m−Ω_Λ) +a⁰⁴Ω_Λ]^−1/2. (1.29)

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Since in cosmology one often uses the redshift dependence instead, we rewrite the comoving distance as

w(z) = c H₀

Z z 0

dz⁰[(1 +z⁰)³Ω_m+ (1 +z⁰)²(1−Ω_m−Ω_Λ) + Ω_Λ]^−1/2. (1.30) The pre-factor is the so-called Hubble distance and has a value of c/H₀ = 2998h⁻¹Mpc.

However, the comoving distance is not directly accessible to observations. Instead, observations measure the angular extent or the luminosity of an object which lead to the definition of the angular diameter distance and luminosity distance, respectively.

Both quantities depend on the comoving distance. Thus, the comoving distance is the most basic distance quantity.

In Euclidean space, the angular extent ∆θ of a source of size l that is located at distance Dfrom the observer is given by

∆θ = l

D, (1.31)

where we used the small-angle approximation. We can use this relation to define the angular diameter distance

D_A = l

∆θ (1.32)

for an arbitrary metric. Suppose the observer is located at a redshift z₁, and the source is lying at a redshift z₂ withz₁ < z₂. From the angular part of the Robertson-Walker metric (see Eq. 1.1) we find then l = a(z2)fK[w(z1, z2)]∆θ. Therefore, the angular diameter distance is given by

D_A(z₁, z₂) = a(z₂)f_K[w(z₁, z₂)]. (1.33) The luminosity distance is especially important for the evidence of dark energy from supernova observations. If we have a source with absolute luminosity L_e at a distance D then an observer at t =t₀ receives a fluxF given by

F = Le

4πD² (1.34)

in Euclidean space. Therefore, we define the luminosity distance as D_L =

L_e 4πF

1/2

. (1.35)

Now we want to calculate the flux for a Robertson-Walker metric. We find that the area of the sphere of the emitted light which reaches the observer at t =t₀ is given by A= 4πa²(t₀)[f_K(w)]². Hence, the observed flux is given by

F = L₀

4π[a₀f_K(w)]² (1.36)

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Due to the redshift of photons, the connection between emitted and received luminosity is

L_e= (1 +z)²L₀. (1.37)

Finally, combining Eqs. (1.35), (1.36) and (1.37) the expression for the luminosity distance is (setting a₀ = 1)

D_L(z) = 1

a(z)f_K[w(z)]. (1.38)

These different distance measures lead in general to different results because the concept of distance is ambiguous in curved space-time in contrast to Euclidean or Minkowski space. For small redshifts (z 1) all distance measures can be described by the local Hubble law

d(z) = v H₀ = c

H₀z (1.39)

because the curvature of the Universe is negligible in this case. The relation between the redshift and the velocity v is due to the linear non-relativistic Doppler effect. The linear behavior between distance and receding velocity was first discovered by Edwin Hubble in 1929 after a long period of measurements of nearby cepheids. It was one of the first hints that the Universe is expanding.

1.2 Energy Composition of the Universe

We review the status of the determination of today’s energy composition of the Universe quantified by the dimensionless density parameters. Ideally, we have many different and/or complementary measurement methods to reduce systematic effects and break parameter degeneracies. This is indeed the case for most parameters and the agreement between different methods strengthens our evidence that we live in a flat Universe dominated by non-baryonic dark matter and dark energy. Finally, we discuss different time evolutions of the expansion of the Universe, so-called world models. With the measurements of today’s cosmological parameters we can infer that the Universe started from a Big Bang singularity. In some parts we follow the discussion in Dodelson (2003).

1.2.1 Radiation Density

The term radiation density refers to the energy density of relativistic particles. Today, the temperature of the Universe dropped significantly compared to the hot plasma after the Big Bang. Hence, the relativistic density today is dominated by massless particles where the most abundant are CMB photons and relic neutrinos. Therefore we define

Ω_r= Ω_γ+ Ω_ν. (1.40)

In the following we determine the amount of both components today.

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Photons

The CMB radiation measured by the COBE³ satellite mission has a perfect Planck spectrum with a temperature T_γ = 2.725±0.002 K (Mather et al. 1999). We can then infer the energy density of photons:

ρ_γ = 2

Z d³p (2π)³

p

exp(p/Tγ)−1 = π²

15T_γ⁴, (1.41)

where we used the fact that photons have two degrees of freedom corresponding to the two polarization states and that the rest mass of the photons is zero, i.e.,E =p

m²+p² = p.

Note that we used natural units in Eq. (1.41) where ~ = c= k_B = 1. Inserting the observed CMB temperature, we can deduce the value of the photon density parameter:

Ω_γ = ρ_γ

ρ_crit = 2.47×10⁻⁵h⁻². (1.42) Note that in addition to the isotropic CMB temperature, the COBE satellite found tiny fluctuations in the angular distribution of the CMB temperature of the order

∆T /T ∝10⁻⁵. We will discuss this observation in Sect. 1.3.3 where we talk about the origin of the CMB radiation.

Neutrinos

In contrast to the CMB radiation, the relic neutrino radiation has not been directly observed because of the small interaction cross sections of these low energy neutrinos.

However, using relatively simple theoretical arguments, we can derive the temperature of the neutrinos from the temperature of the CMB photons.

In the early Universe neutrinos were kept in equilibrium with electrons by weak interactions. Hence, they are part of a thermal bath where the particles share the same temperature. At an energy slightly above 1MeV the neutrino interaction rates become so small that they decouple from the thermal bath. Shortly thereafter, at the energy of E = 2me '1MeV, the pair production of electrons and positronsγγ →e⁺e⁻ stops and because neutrinos are absent from the thermal bath the annihilation energy of the electron-positron pairs is only transferred to the photons. As a consequence, photons are hotter than the neutrinos. Using the conservation of the entropy density we find

T_ν T_γ =

4 11

1/3

, (1.43)

where Tν and Tγ denote the neutrino and photon temperature, respectively. Plugging this result into Eq. (1.41) leads to

ρν = 37 8

4 11

4/3

ργ, (1.44)

3Cosmic Background Explorer.

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where the factor 7/8 is due to the fact that neutrinos obey Fermi-Dirac statistics in contrast to the photons which obey Bose-Einstein statistics. In addition, considering that there are 3 generations of neutrinos and anti-neutrinos, they have 6 degrees of freedom⁴ compared to the 2 degrees of freedom of photons. This results in the factor of 3 for the density of neutrinos. Finally, we find for the density parameter of neutrinos

Ω_ν = 1.68×10⁻⁵h⁻². (1.45)

Combining the results from photons (1.42) and neutrinos (1.45), we get Ω_r = 4.15× 10⁻⁵h⁻².

We have to keep in mind that the derivation of this result is only valid for massless neutrinos. Recently, however, there is strong evidence for massive neutrinos from atmospheric (Fukuda et al. 1998) and solar neutrino oscillation experiments (Ahmad et al. 2001). The result of the experiments is that at least one of the neutrinos has a rest mass above m_ν = 0.05eV. The effect of massive neutrinos is that at high temperatures they behave relativistically, whereas at low temperatures there is a turnover in the density evolution, where neutrinos act as non-relativistic matter with a scalingρ_ν ∝a⁻³. This behavior makes massive neutrinos a natural candidate for non-baryonic dark matter (see also next section below). However, intensive studies on structure formation using numerical simulations show that relativistic neutrinos (so-called hot dark matter) cannot reproduce the statistical properties of the large-scale structure we observe today.

Neutrinos would “wash out” perturbations of the background cosmology and significantly reduce the formation of bound objects like galaxies and galaxy clusters. Hence, at least partly, dark matter must be composed of an unknown type of matter which is able to reproduce the statistical properties of the observed large-scale structure.

1.2.2 Matter Density

The matter density is composed of non-relativistic particles. Today, this includes ordinary baryonic matter and non-baryonic dark matter (plus a small contribution from massive neutrinos), and thus the density parameter is Ω_m = Ω_b+ Ω_dm.

Baryon Density

There are many established ways to infer the baryon density of the Universe. The simplest method is to determine the amount of gas in groups and clusters of galaxies.

Note that the contribution from the intergalactic medium is larger than the contribution from individual stars. Another method are measurements of spectra of distant quasars, where the amount of absorbed gas is a measure of the intervening hydrogen, and thus an estimate of the baryon density. However, the two most precise predictions come from measurements of the CMB temperature fluctuations and of the abundance of primordial elements. Changing the baryon density of the Universe leaves characteristic

4Each neutrino has one spin degree of freedom (g_ν = 1).

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imprints in the temperature fluctuations of the CMB radiation making it one of the easiest parameters to extract.

The amount of primordial deuterium predicted from nucleosynthesis is very sensitive to the assumed baryon-to-photon ratio. Thus, measurements of primordial deuterium can be compared to model predictions to infer the photon-to-baryon ratio. Since we know the photon density very well from the CMB radiation, we obtain an estimate for the baryon density. We can measure the deuterium abundance by determining the absorption of light from distant QSOs by intervening neutral hydrogen systems (Burles

& Tytler 1998). The main absorption feature is the Lyα-transition, which is slightly different for hydrogen and deuterium due to the different masses of the nuclei. Hence, in the absorption spectrum, deuterium leaves a clear feature which is shifted relative to the hydrogen spectrum and is less damped because of the small amount of deuterium.

Remarkably, the result from the presented methods agree and yield Ω_bh² ≈0.02. For a fiducial value of h= 0.7, we obtain Ω_b≈0.04.

Dark Matter Density

The presented methods to obtain the baryon density all involve the interactions between matter and radiation. In contrast, measurements that are sensitive to the gravitational field of bound systems only depend on the total mass of the system. Surprisingly, there is very strong evidence that both mass estimates disagree leading to the proposal of an unknown form of matter that does not interact with electromagnetic radiation. In the following we present different probes that measure the total mass.

One of the first probes to infer an estimate was the measurement of the mass-to- light ratio of bound systems (e.g., Tinker et al. 2005). For small to intermediate size systems, the mass-to-light ratio is an increasing function of the length scale. However, on the largest scales corresponding to galaxy clusters and superclusters the mass-to-light ratio approaches a constant value. Hence, if we assume that the matter content in superclusters is representative of the matter content of the Universe, we can infer the total mass density Ωm.

More recent methods involve observations of the distribution of galaxies which depends on the matter density of the Universe (Cole et al. 2005). A very similar method is the determination of the peculiar velocity field which can be simply related to the density field by using the continuity equation (Hawkins et al. 2003). Finally, the distribution of the temperature fluctuations of the CMB radiation is sensitive to the combination Ω_mh².

Another approach is to use methods that depend on the ratio Ω_b/Ω_m. Then using the result of the baryon density of the previous section, we can deduce the total matter density. An example are galaxy clusters, where most of the baryonic mass is situated in the intercluster medium in the form of hot gas. Measurements of the X-ray emission of the hot gas or the scattering of CMB photons of the electrons in the plasma (Sunyaev-Zel’dovich effect) are sensitive to the amount of gas relative to the total mass (LaRoque et al. 2006). Clusters collapse from a large volume of the order of 1000Mpc

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and thus their composition should reflect the baryon-dark matter ratio of the Universe.

Furthermore, baryonic acoustic oscillations in the baryon-photon plasma in the early Universe leave characteristic imprints in the galaxy distribution that can be used to infer the density ratio (Eisenstein et al. 2005).

This is only an excerpt of the various number of cosmological probes. Most notably, we left out the discussion of the cosmic shear power spectrum which is introduced in Sect. 5.2. All different probes obtain Ω_m ' 0.3 for the total matter density of the Universe. Hence, this result requires in addition to the baryonic matter another form of matter that interacts only gravitationally, dubbed as dark matter. Furthermore, compared to the result of the baryon density in the previous section, this unknown form of matter clearly dominates the total matter density.

1.2.3 Dark Energy

The recent Universe undergoes a phase of accelerated expansion rather than a slowdown which can be explained by a recent domination of a new energy component, termed dark energy. We show the evidence for dark energy and give results to its contribution to the energy budget of the Universe.

Measurements of the CMB temperature fluctuations determine at high accuracy that the density of the Universe is very close to critical implying that the Universe is flat.

However, the total matter density only contributes roughly a third to the critical density and the amount of the radiation density is negligible today. Hence, to achieve the observed flatness, where Ω_tot = 1 (see Eq. 1.24), we need to propose another dominating energy form that fulfills Ω_Λ= 1−Ω_m.

The other evidence comes from cosmological probes that are sensitive to distance- redshift relations (see Eqs. 1.33 and 1.38) and/or the Hubble rate H(z) (see Eq. 1.25) because they directly depend on the amount of Ω_Λ. An example is the observation of SNIa as standard candles to obtain the distance-redshift relation of the luminosity distance. In particular, the results suggest that the luminosity distance is best fitted by a theoretical model with a dominant contribution from a cosmological constant. In contrast anEinstein-de Sitter model⁵, where Ω_m= 1 and Ω_Λ= 0, is strongly disfavored by the data. Especially the large amount of dark energy came as a big surprise to most cosmologists. However, there were also still doubts about the evidence because of the partly unknown amount of systematics in the measurements, for example in the understanding of supernova explosions. From then on, we have collected ever growing evidence on dark energy as a result of the availability of larger data sets and improvements in the modeling of systematics. In addition, complementary probes like gravitational lensing caused by the large-scale structure (cosmic shear), the CMB temperature fluctuations, observations of galaxy clustering, baryonic acoustic oscillations, cluster mass function, etc., also need dark energy. Hence, the new evidence put the recent dark energy domination on a firm foundation, and the different probes find Ω_Λ≈0.7.

5For a long time this was the standard cosmological model of the Universe.

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On the theoretical side there are fundamental problems to find a model that predicts the observed amount of dark energy. The first idea is to assume that dark energy corresponds to the energy of the vacuum of the Universe. However, standard quantum field theory predictions are about 10¹²¹ orders of magnitude larger than the observed value. This is known as the famous cosmological constant problem. These difficulties lead to the proposal of a time-varying dark energy component, which is described by so-called quintessence models. Nevertheless, the problems to produce a theory that solves the cosmological constant problem and is falsifiable remains and a lot of effort is put in new theoretical models and in observations that obtain improved constraints on the amount and the EOS of dark energy. In particular, the main goal is to determine the EOS of dark energy at percentage-level accuracy to be able to distinguish between cosmological constant and dynamical (time-varying) dark energy models.

Basically there are three types of theoretical models to explain dark energy. The first idea is to change the right-hand side of the Einstein equation (1.9), which is equivalent to changing the energy density of the Universe. On the other hand, we can also change the left-hand side side of the Einstein equation which corresponds to a modification of gravity. Both solutions have their own difficulties. For example a modification of gravity still needs to converge to Einstein’s theory on small scales because it is very well tested on solar system scales. The third approach to explain dark energy are the so-called backreaction models, where inhomogeneities in the large-scale structure lead to a breakdown of the cosmological principle. In this scenario we have to modify the Friedmann equation and need to introduce averaging schemes over the inhomogeneities of the large-scale structure. For comprehensive reviews of the work on theoretical models and future experiments we refer to the report of the dark energy task force (Albrecht et al. 2006) and the reviews by Copeland (2007) and Frieman et al. (2008).

1.2.4 Cosmological World Models

With the knowledge of the energy composition, we can now constrain the past and future global evolution of the Universe and provide evidence for the presence of a Big Bang. This problem involves solving the first Friedmann equation (1.25) for the scale factor a as a function of cosmic time t. In general this can be done only numerically.

Today we can neglect the radiation density parameter due to its smallness (Ωr'10⁻⁵).

Qualitatively, we have then an interplay between the attractive gravitational force acting on matter and the repulsive or attracting⁶ force of the cosmological constant. There are three types of world models:

• an eternally expanding Universe,

• a Universe that will recollapse in the future, i.e., the expansion of the Universe stops and turns into a contraction,

6We are also considering negative values for Ω_Λ, in which case dark energy is an attractive force.

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No Big Bang

1 2

0 1 2 3

expands forever

-1 0 1 2 3

2 3

closed recollapses eventually

Supernovae

CMB Boomerang

Maxima

Clusters

mass density vacuum energy density (cosmological constant)

open flat

SNAP

Target Statistical Uncertainty

Figure 1.1: Ω_m-Ω_Λ-plane with confidence regions for three cosmological probes: SNIa, CMB temperature fluctuations and galaxy clustering. Each individual probe show large degeneracies between the two parameters and is thus not capable to give tight constraints on the cosmological model. However, the combination of all three observations clearly favors a flat ΛCDM model with a dominant contribution from a cosmological constant. In addition, the diagram is separated into regions corresponding to different geometries of the Universe. Taken from Aldering et al. (2002).

• a bouncing Universe, i.e., a Universe that started without a Big Bang singularity at the scale factor a then contracted due to matter domination and turned over to expansion at a <1. A subclass of this model is theloitering Universe which is the critical case between bouncing and eternally expanding Universe. It lingers for a long time at a constant redshift zmax when the influence of the Λ-term takes over and yields an expanding Universe.

To illustrate the dependence of these different world models on cosmological parameters, we show in Fig. 1.1 the Ωm-ΩΛ-plane. One can read off the following results:

• A negative Λ always implies recollapse, which is clear because Λ supports in this case the gravitational attraction of matter.

• For Ω_Λ >0 and Ω_m<1 the Universe always expands to infinity.

• For Ω_mΩ_Λ the recollapse is still possible with a positive Ω_Λ <1,

• For ΩΛ >1 it is possible to find a bouncing Universe depending on the matter content of the Universe.

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Bouncing models can be ruled out by observations of high-redshift objects. Depending on the matter content Ω_m, one can calculate a maximal redshift that is obtained in this model. For example, a conservative lower limit of Ωm≥0.1 rules out a bouncing Universe once objects are seen at redshifts beyond z = 2. This is indeed the case as observations of high-redshift quasars beyond z >6 show. Hence, we conclude that the Universe started from a Big Bang singularity. In addition, the combination of results from SNIa, galaxy clusters and CMB experiments in Fig. 1.1 show that we are living in a flat, dark energy dominated Universe. Hence, the scenario that the Universe recollapses in the future with a Big Crunch is strongly ruled out by the data.

1.3 The Cosmological Standard Model and Extensions

The cosmological standard model is based on three important observations: the expansion of the Universe discovered by Hubble, primordial nucleosynthesis and the cosmic microwave background radiation. After reviewing the important epochs in the history of the Universe, we present the predictions of primordial nucleosynthesis and of the CMB radiation released at recombination. We conclude this section with an important extension of the standard model: the inflationary phase of the very early Universe.

1.3.1 History of the Universe

The history of the Universe begins at the Big Bang singularity. Very shortly thereafter it undergoes an accelerated expansion due to inflation. In addition, inflation amplifies the quantum fluctuations to build the seeds of structure formation. The early Universe is composed of a hot and dense plasma, where particles are maintained in thermal equilibrium by the rapid rate of particle interactions. On the other hand, the expansion of the Universe results in a subsequent cooling which leads to a reduction in the particle interaction rates. As a consequence, particles for which the Hubble expansion is larger than their corresponding interaction rate are not able to maintain equilibrium anymore and freeze out. The next important epoch is the process of nucleosynthesis, where the temperature is low enough that neutrons and protons can form deuterium which leads in the end to the formation of helium nuclei. Going back to the solution of the energy density for matter and radiation in Eq. (1.21), we note that there is a time in the past where both components are of the same size which is known asmatter-radiation equality.

Combining the result of Sect. 1.2.1 with the definition of the matter density, we find a_eq = 4.15×10⁻⁵Ω⁻¹_m h⁻². (1.46) Hence, the larger the mass density the earlier the time of matter-radiation equality.

At around 300 000 years (z '1100) after the Big Bang the Universe cooled enough to allow electrons and protons to combine to build neutral hydrogen⁷. This epoch is called

7The recombination of helium atoms occurs at higher redshifts.

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Figure 1.2:Distribution of galaxies observed in the Two Degree Field Galaxy Redshift Survey (2dF) (Colless et al. 2001). As can be seen in the figure, the survey probes in this data release the galaxy distribution up to redshifts of z≈0.2.

recombination and is the origin of the CMB radiation as the photons can now travel freely through the Universe. From then on until the formation of the first objects we are unable to probe the structure of the Universe and this epoch is therefore called the dark ages. The only indirect observable is the light emitted from the 21 centimeter transition of neutral hydrogen⁸. During the dark ages the hierarchical growth of structure due to selfgravity leads to the first giant metal-poor stars, the so-called population III stars, or AGNs. Subsequent explosions of these first giant stars lead then to an ionization of the surrounding gas. These ionized regions begin then to grow and overlap. The metal enrichment of the explosions of population III stars lead to the possibility that population II stars can be formed which also contribute to the ionization of the Universe.

Observations of distant quasars and of the CMB radiation show the reionization takes place around z '6–15 in the history of the Universe (see the review Barkana & Loeb 2007). Ongoing structure formation results in the large-scale structure we observe today composed of galaxies, clusters, voids, filaments etc. (see results of the galaxy distribution of the 2dF in Fig. 1.2). Only very recently the Universe changed from a slowdown due to the matter domination to an acceleration of the expansion due to dark energy domination.

8The 21 centimeter line comes from the transition of the two hyperfine lines in the 1s ground state.

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1.3.2 Primordial Nucleosynthesis

The theory of Big Bang nucleosynthesis (BBN) mainly predicts the fraction of helium nuclei produced in the early Universe through the following reaction chain

p+n→D +γ , D + D →n+³He, ³He + D→p+⁴He. (1.47) However, the number density of baryons is much smaller than the number density of photons and thus any time a nucleus is formed in a reaction it is destroyed by a high-energy photon. The formed nuclei can only remain stable when the temperature of the Universe drops below a critical temperature. The first reaction of the chain (1.47) indicates that the neutron-to-proton ratio is a key quantity for the formation of helium.

The following weak interactions keep neutrons and protons in equilibrium until T ≈1 MeV:

p+ ¯ν ↔n+e⁺, p+e⁻↔n+ν , n→p+e⁻+ ¯ν . (1.48) In equilibrium the neutron-to-proton ratio in the nonrelativistic limit is given by

n⁽⁰⁾n

n⁽⁰⁾p

= e^−Q/(k^B^T⁾, (1.49)

where Q = (m_n−m_p)c² = 1.293MeV is the energy of the mass difference between neutrons and protons. Hence, at high temperatures the number density of neutrons is equal to the number density of protons. On the other hand, as the temperature drops below 1MeV, the neutron fraction gets reduced. Considering only the equilibrium case, where weak interactions are completely efficient, the neutron fraction would drop to zero. However, at temperatures below 1MeV we have to consider out-of-equilibrium processes which involves solving the Boltzmann equation for the reactions in Eq. (1.48) to obtain the neutron-to-proton ratio (see Dodelson 2003). The (integrated) Boltzmann equation describes the fact that the rate of change in the abundance of a given particle is equal to the difference between the production and elimination rates of that particle. At temperatures below 0.1MeVthe decay of neutrons (n →p+e⁻+ν) and the production¯ of deuterium (n+p → D +γ) become important resulting in a strong reduction of neutrons. To quantify this reduction, we define the ratio

X_n≡ n_n

n_n+n_p . (1.50)

Numerical solutions of the Boltzmann equation show that at a temperature around T_nuc = 0.07MeV the formation of deuterium and very shortly thereafter of helium begins, and the fraction is reduced toXn(Tnuc) = 0.11. All neutrons present combine to the stable nucleus of ⁴He which has the highest binding energy of light nuclei. Since each helium nuclei is composed of two neutrons, we haven⁴_He =n_n/2. The ratio of ⁴He to the total baryon density is given by (Cyburt et al. 2005)

Y = 4n⁴_He

4n⁴_He+n_H = 2n_n

n_n+n_p = 2X_n= 0.22. (1.51)

Studying Galaxy-Galaxy Lensing and Higher-Order Galaxy-Mass Correlations Using the Halo Model